Xenharmonic Wiki

Canopic clan: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
recat
m FloraC moved page Canopus clan to Canopic clan
 
(59 intermediate revisions by 11 users not shown)
Line 1: Line 1:
Temperaments of the '''Mirkwai clan''' temper out the ''mirkwai comma'' ({{Monzo| 0 3 4 -5 }} = 16875/16807), a no-twos comma. Members of the clan include grendel, kwai, pluto and mirkat considered below, as well as miracle, octokaidecal, nusecond, semisept and bohpier, considered elsewhere.
{{Technical data page}}
The '''canopic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[canopic comma]] ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.  


=Grendel=
== Canopus ==
Commas: 6144/6125, 16875/16807
{{Main| Canopus }}


[[POTE_tuning|POTE generator]]: 386.863
[[Subgroup]]: 3.5.7


Map: [<1 9 2 7|, <0 -23 1 -13|]
[[Comma list]]: 16875/16807


Wedgie: <<23 -1 13 -55 -44 33||
{{Mapping|legend=2| 1 -2 -1 | 0 5 4 }}
: mapping generators: ~3, ~15/7


EDOs: 31, 90, 121, 152, 335
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1901.7826{{c}}, ~15/7 = 1317.8771{{c}}
: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~15/7 = 1317.9686{{c}}
: error map: {{val| 0.000 -0.381 +1.093 }}


==11-limit==
[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]
Commas: 540/539, 1375/1372, 5632/5625


[[POTE_tuning|POTE generator]]: 386.856
[[Badness]] (Sintel): 0.0996


Map: [<1 9 2 7 17|, <0 -23 1 -13 -42|]
=== Overview to extensions ===
The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.
* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]
* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]


EDOs: 31, 90, 121, 152, 335, 487
The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Kwai tempers out [[5120/5103]], splitting the generator in ten. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen.


==13-limit==
Members of the clan discussed elsewhere are:
Commas: 352/351, 540/539, 625/624, 1375/1372
* ''[[Kwai]]'' (+5120/5103) → [[Hemifamity temperaments #Kwai|Hemifamity temperaments]]
* ''[[Octokaidecal]]'' (+28/27 or 50/49) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Meantritone]]'' (+81/80) → [[Meantone family #Meantritone|Meantone family]]
* ''[[Quanharuk]]'' (+32805/32768) → [[Schismatic family #Quanharuk|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Bohpier]]'' (+245/243) → [[Sensamagic clan #Bohpier|Sensamagic clan]]
* ''[[Subsedia]]'' (+65536/64827) → [[Buzzardsmic clan #Subsedia|Buzzardsmic clan]]
* ''[[Semisept]]'' (+1728/1715 or 3136/3125) → [[Hemimean clan #Semisept|Hemimean clan]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Quinmage]]'' (+3125/3072) → [[Magic family #Quinmage|Magic family]]
* ''[[Familia]]'' (+1600000/1594323) → [[Amity family #Familia|Amity family]]
* [[Sqrtphi]] (+15625/15552) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Rainwell]]'' (+2100875/2097152) → [[Semicomma family #Rainwell|Semicomma family]]
* ''[[Quintiquart]]'' (+390625000/387420489) → [[Quartonic family #Quintiquart|Quartonic family]]


[[POTE_tuning|POTE generator]]: 386.826
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #Canopus]].


Map: [<1 9 2 7 17 -5|, <0 -23 1 -13 -42 27|]
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, hemiseptisix, browser, and grazer, in the order of increasing [[badness]].


EDOs: 31, 121, 152, 425
== Mirkat ==
Mirkat tempers out 19683/19600, the [[cataharry comma]], as well as 250047/250000, the [[landscape comma]], and may be described as the {{nowrap| 72 & 111 }} temperament with a [[ploidacot]] signature of triploid alpha-hexacot.


==17-limit==
[[Subgroup]]: 2.3.5.7
Commas: 256/255, 352/351, 625/624, 715/714, 1275/1274


[[POTE_tuning|POTE generator]]: 386.812
[[Comma list]]: 16875/16807, 19683/19600


Map: [<1 9 2 7 17 -5 -3|, <0 -23 1 -13 -42 27 22|]
{{Mapping|legend=1| 3 2 1 2 | 0 6 13 14 }}
: mapping generators: ~63/50, ~10/9


EDOs: 31, 121, 273
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.0277{{c}}, ~10/9 = 183.5515{{c}}
: [[error map]]: {{val| +0.083 -0.591 -0.117 +0.950 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5470{{c}}
: error map: {{val| 0.000 -0.673 -0.203 +0.831 }}


==19-limit==
{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}
Commas: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714


[[POTE_tuning|POTE generator]]: 386.819
[[Badness]] (Sintel): 1.50


Map: [<1 9 2 7 17 -5 -3|, <0 -23 1 -13 -42 27 22|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 31, 121, 152, 273
Comma list: 540/539, 1375/1372, 8019/8000


=Kwai=
Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}
Commas: 5120/5103, 16875/16807


[[POTE_tuning|POTE generator]]: 702.616
Optimal tunings:  
* WE: ~63/50 = 400.0463{{c}}, ~10/9 = 183.5496{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5391{{c}}


Map: [<1 0 -50 -40|, <0 1 33 27|]
{{Optimal ET sequence|legend=0| 39d, 72, 111, 183, 255 }}


Wedgie: <<1 33 27 50 40 -30||
Badness (Sintel): 0.731


EDOs: 41, 111, 152, 345, 497
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 351/350, 540/539, 676/675, 1375/1372
Commas: 540/539, 1375/1372, 16384/16335


[[POTE_tuning|POTE generator]]: 702.623
Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}


Map: [<1 0 -50 -40 32|, <0 1 33 27 -18|]
Optimal tunings:  
* WE: ~63/50 = 400.0245{{c}}, ~10/9 = 183.5885{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~10/9 = 183.5825{{c}}


EDOs: 29, 41, 111, 152
{{Optimal ET sequence|legend=0| 39df, 72, 111, 183 }}


==13-limit==
Badness (Sintel): 0.770
Commas: 352/351, 540/539, 729/728, 1375/1372


[[POTE_tuning|POTE generator]]: 702.644
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [<1 0 -50 -40 32 27|, <0 1 33 27 -18 -21|]
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714


EDOs: 29, 41, 111, 152
Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}


==17-limit==
Optimal tunings:
Commas: 256/255, 352/351, 540/539, 715/714, 1089/1088
* WE: ~34/27 = 400.0257{{c}}, ~10/9 = 183.5906{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~10/9 = 183.5843{{c}}


[[POTE_tuning|POTE generator]]: 702.660
{{Optimal ET sequence|legend=0| 39dfg, 72, 111, 183 }}


Map: [<1 0 -50 -40 32 27 58|, <0 1 33 27 -18 -21 -34|]
Badness (Sintel): 0.600


EDOs: 29, 41, 111, 152, 263
== Eris ==
Eris tempers out 65625/65536, the [[horwell comma]], and may be described as the {{nowrap| 31 & 224 }} temperament. The [[2.5.7-subgroup|2.5.7 subgroup]] restriction of this temperament is [[exodia]].


==19-limit==
[[Subgroup]]: 2.3.5.7
Commas: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


[[POTE_tuning|POTE generator]]: 702.657
[[Comma list]]: 16875/16807, 65625/65536


Map: [<1 0 -50 -40 32 27 58 -56|, <0 1 33 27 -18 -21 -34 38|]
{{Mapping|legend=1| 1 -19 8 -5 | 0 29 -8 11 }}
: mapping generators: ~2, ~49/30


EDOs: 29, 41, 111, 152, 263
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0256{{c}}, ~49/30 = 851.8023{{c}}
: [[error map]]: {{val| +0.026 -0.173 -0.528 +0.872 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/30 = 851.7845{{c}}
: error map: {{val| 0.000 -0.204 -0.590 +0.804 }}


=Pluto=
{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224 }}
Commas: 4000/3969, 10976/10935


[[POTE_tuning|POTE generator]]: 585.147
[[Badness]] (Sintel): 1.89


Map: [<1 5 15 15|, <0 -7 -26 -25|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: <<7 26 25 25 20 -15||
Comma list: 540/539, 1375/1372, 65625/65536


EDOs: 39, 41, 80, 121, 404
Mapping: {{mapping| 1 -19 8 -5 -37 | 0 29 -8 11 57 }}


==11-limit==
Optimal tunings:
Commas: 540/539, 896/891, 1375/1372
* WE: ~2 = 1200.0218{{c}}, ~18/11 = 851.7963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 851.7812{{c}}


[[POTE_tuning|POTE generator]]: 585.114
{{Optimal ET sequence|legend=0| 31, …, 193, 224, 703, 927d }}


Map: [<1 5 15 15 2|, <0 -7 -26 -25 3|]
Badness (Sintel): 0.913


EDOs: 39, 41, 80, 121
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
Commas: 325/324, 352/351, 364/363, 540/539


[[POTE_tuning|POTE generator]]: 585.123
Mapping: {{mapping| 1 -19 8 -5 -37 47 | 0 29 -8 11 57 -61 }}


Map: [<1 5 15 15 2 -8|, <0 -7 -26 -25 3 24|]
Optimal tuning:  
* WE ~2 = 1199.9623{{c}}, ~18/11 = 851.7598{{c}}
* CWE ~2 = 1200.0000{{c}}, ~18/11 = 851.7865{{c}}


EDOs: 39, 41, 80, 121
{{Optimal ET sequence|legend=0| 31, 193, 224 }}


==17-limit==
Badness (Sintel): 1.04
Commas: 256/255, 325/324, 352/351, 364/363, 540/539


[[POTE_tuning|POTE generator]]: 585.116
== Subsemifourth ==
[[Subgroup]]: 2.3.5.7


Map: [<1 5 15 15 2 -8 -12|, <0 -7 -26 -25 3 24 33|]
[[Comma list]]: 16875/16807, 26873856/26796875


EDOs: 39, 41, 80, 121
{{Mapping|legend=1| 1 -8 -4 -8 | 0 47 31 53 }}
: mapping generators: ~2, ~144/125


==19-limit==
[[Optimal tuning]]s:
Commas: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
* [[WE]]: ~2 = 1199.9182{{c}}, ~144/125 = 244.7020{{c}}
: [[error map]]: {{val| -0.082 -0.305 -0.223 +1.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/125 = 244.7172{{c}}
: error map: {{val| 0.000 -0.248 -0.082 +1.184 }}


[[POTE_tuning|POTE generator]]: 585.109
{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407 }}


Map: [<1 5 15 15 2 -8 -12 14|, <0 -7 -26 -25 3 24 33 -20|]
[[Badness]] (Sintel): 3.42


EDOs: 39, 41, 80, 121, 564, 685
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Orcus==
Comma list: 540/539, 1375/1372, 234375/234256
Commas: 144/143, 196/195, 275/273, 896/891


POTE generator: ~7/5 = 585.111
Mapping: {{mapping| 1 -8 -4 -8 -10 | 0 47 31 53 66 }}


Map: [<1 5 15 15 2 12|, <0 -7 -26 -25 3 -17|]
Optimal tunings:  
* WE: ~2 = 1199.9229{{c}}, ~121/105 = 244.7033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/105 = 244.7175{{c}}


EDOs: 41, 80f, 121f
{{Optimal ET sequence|legend=0| 49, 103, 152, 255, 407 }}


Badness: 0.0334
Badness (Sintel): 1.13


=Plutino=
=== 13-limit ===
Commas: 100/99, 245/242, 16875/16807
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 585.283
Comma list: 540/539, 847/845, 1375/1372, 1575/1573


Map: [<1 5 15 15 22|, <0 -7 -26 -25 -38|]
Mapping: {{mapping| 1 -8 -4 -8 -10 -12 | 0 0 47 31 53 66 77 }}


EDOs: 41, 123c
Optimal tunings:  
* WE: ~2 = 1199.9003{{c}}, ~15/13 = 244.6932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/13 = 244.7116{{c}}


Badness: 0.0580
{{Optimal ET sequence|legend=0| 49f, 103, 152f, 255, 407f }}


==13-limit==
Badness (Sintel): 1.17
Commas: 100/99, 196/195, 245/242, 729/728


POTE generator: ~7/5 = 585.232
== Septendesemi ==
Septendesemi tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma) in the 7-limit, and may be described as the {{nowrap| 80 & 103 }} temperament. [[183edo]] provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi. Septendesemi was named by [[Xenllium]] in 2021; the name ''septendesemi'' refers to a septendecimal semitone ([[17/16]]).  


Map: [<1 5 15 15 22 12|, <0 -7 -26 -25 -38 -17|]
[[Subgroup]]: 2.3.5.7


EDOs: 41, 121ef
[[Comma list]]: 16875/16807, 1959552/1953125


Badness: 0.0402
{{Mapping|legend=1| 1 -2 -1 -2 | 0 41 38 55 }}
: mapping generators: ~2, ~343/324


=Eris=
[[Optimal tuning]]s:
Commas: 16875/16807, 65625/65536
* [[WE]]: ~2 = 1199.8649{{c}}, ~343/324 = 104.9046{{c}}
: [[error map]]: {{val| -0.135 -0.597 +0.195 +1.196 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/324 = 104.9134{{c}}
: error map: {{val| 0.000 -0.506 +0.395 +1.410 }}


POTE generator: ~60/49 = 348.216
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


Map: [<1 10 0 6|, <0 -29 8 -11|]
[[Badness]] (Sintel): 3.71


Wedgie: <<29 -8 11 -80 -64 48}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 31, 131, 162, 193, 224, 1823cd, 2271cd
Comma list: 540/539, 1375/1372, 43923/43750


Badness: 0.0747
Mapping: {{mapping| 1 -2 -1 -2 -1 | 0 41 38 55 51 }}


==11-limit==
Optimal tunings:
Commas: 540/539, 1375/1372, 65625/65536
* WE: ~2 = 1199.9327{{c}}, ~35/33 = 104.9100{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9144{{c}}


POTE generator: ~11/9 = 348.219
{{Optimal ET sequence|legend=0| 80, 103, 183 }}


Map: [<1 10 0 6 20|, <0 -29 8 -11 -57|]
Badness (Sintel): 1.37


EDOs: 31, 193, 224, 703, 927d, 1151cd
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0276
Comma list: 351/350, 540/539, 1375/1372, 4225/4224


=Mirkat=
Mapping: {{mapping| 1 -2 -1 -2 -1 3 | 0 41 38 55 51 8 }}
Commas: 16875/16807, 19683/19600


POTE generator: ~10/9 = 183.539
Optimal tunings:  
* WE: ~2 = 1200.1082{{c}}, ~35/33 = 104.9170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 104.9094{{c}}


Map: [<3 2 1 2|, <0 6 13 14|]
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Wedgie: <<18 39 42 20 16 -12||
Badness (Sintel): 1.15


EDOs: 39d, 72, 111, 183, 255
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0594
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224


==11-limit==
Mapping: {{mapping| 1 -2 -1 -2 -1 3 4 | 0 41 38 55 51 8 1 }}
Commas: 540/539, 1375/1372, 8019/8000


POTE generator: ~10/9 = 183.528
Optimal tunings:  
* WE: ~2 = 1200.0758{{c}}, ~17/16 = 104.9158{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/16 = 104.9101{{c}}


Map: [<3 2 1 2 9|, <0 6 13 14 3|]
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~17/16 = 104.909{{c}}


EDOs: 39d, 72, 111, 183, 255
{{Optimal ET sequence|legend=0| 80, 103, 183, 469f }}


Badness: 0.0221
Badness (Sintel): 1.03


==13-limit==
== Gaster ==
Commas: 351/350, 540/539, 676/675, 1375/1372
: ''For the 5-limit version, see [[Very high accuracy temperaments #Gaster]].''
{{Main| Gaster temperament }}


POTE generator: ~10/9 = 183.577
Gaster tempers out {{monzo| -70 72 -19 }} in the 5-limit, mirkwai comma (16875/16807) and [[scheme comma]] (14348907/14336000) in the 7-limit, and may be described as the {{nowrap| 111 & 113 }} temperament.  


Map: [<3 2 1 2 9 1|, <0 6 13 14 3 22|]
It was named by [[Xenllium]] in 2022; the word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.


EDOs: 39df, 72, 111, 183, 255f
[[Subgroup]]: 2.3.5.7


Badness: 0.0186
[[Comma list]]: 16875/16807, 14348907/14336000


==17-limit==
{{Mapping|legend=1| 1 -8 -34 -32 | 0 19 72 69 }}
Commas: 351/350, 442/441, 540/539, 561/560, 715/714
: mapping generators: ~2, ~567/400


POTE generator: ~10/9 = 183.579
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9920{{c}}, ~567/400 = 605.3546{{c}}
: [[error map]]: {{val| -0.008 -0.152 -0.506 +0.902 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~567/400 = 605.3586{{c}}
: error map: {{val| 0.000 -0.142 -0.497 +0.915 }}


Map: [<3 2 1 2 9 1 4|, <0 6 13 14 3 22 18|]
{{Optimal ET sequence|legend=1| 111, 224 }}


EDOs: 39dfg, 72, 111, 183, 255f
[[Badness]] (Sintel): 3.91


=Subsemifourth=
=== 11-limit ===
Commas: 16875/16807, 26873856/26796875
Subgroup: 2.3.5.7.11


POTE generator: ~144/125 = 244.719
Comma list: 540/539, 1375/1372, 14348907/14336000


Map: [<1 39 27 45|, <0 -47 -31 -53|]
Mapping: {{mapping| 1 -8 -34 -32 8 | 0 19 72 69 -9 }}


Wedgie: <<47 31 53 -60 -48 36||
Optimal tunings:  
* WE: ~2 = 1199.9387{{c}}, ~363/256 = 605.3300{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~363/256 = 605.3603{{c}}


EDOs: 49, 103, 152, 255, 407
{{Optimal ET sequence|legend=0| 111, 224, 783d }}


Badness: 0.1352
Badness (Sintel): 1.79


==11-limit==
=== 13-limit ===
Commas: 540/539, 1375/1372, 234375/234256
Subgroup: 2.3.5.7.11.13


POTE generator: ~121/105 = 244.719
Comma list: 540/539, 729/728, 1375/1372, 2200/2197


Map: [<1 39 27 45 56|, <0 -47 -31 -53 -66|]
Mapping: {{mapping| 1 -8 -34 -32 8 -19 | 0 19 72 69 -9 45 }}


EDOs: 49, 103, 152, 255, 407, 966d
Optimal tunings:  
* WE: ~2 = 1199.9154{{c}}, ~78/55 = 605.3183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/55 = 605.3601{{c}}


Badness: 0.0343
{{Optimal ET sequence|legend=0| 111, 224, 783df }}


==13-limit==
Badness (Sintel): 1.03
Commas: 540/539, 847/845, 1375/1372, 1575/1573


POTE generator: ~15/13 = 244.714
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [<1 39 27 45 56 65|, <0 -47 -31 -53 -66 -77|]
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197


EDOs: 103, 152f, 255, 407f, 662df
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 | 0 19 72 69 -9 45 20 }}


Badness: 0.0284
Optimal tunings:  
* WE: ~2 = 1199.8076{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3626{{c}}


[[Category:Theory]]
{{Optimal ET sequence|legend=0| 111, 224, 559dgg }}
[[Category:Temperament clan]]
 
[[Category:Mirkwai]]
Badness (Sintel): 1.09
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 | 0 19 72 69 -9 45 20 56 }}
 
Optimal tunings:
* WE: ~2 = 1199.7542{{c}}, ~17/12 = 605.2674{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3613{{c}}
 
{{Optimal ET sequence|legend=0| 111, 224 }}
 
Badness (Sintel): 1.12
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 | 0 19 72 69 -9 45 20 56 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.8733{{c}}, ~17/12 = 605.2946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3575{{c}}
 
{{Optimal ET sequence|legend=0| 111, 224 }}
 
Badness (Sintel): 1.26
 
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 | 0 19 72 69 -9 45 20 56 5 -32 }}
 
Optimal tunings:
* WE: ~2 = 1199.9442{{c}}, ~17/12 = 605.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3541{{c}}
 
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
 
Badness (Sintel): 1.41
 
=== 31-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 | 0 19 72 69 -9 45 20 56 5 -32 -10 }}
 
Optimal tunings:
* WE: ~2 = 1199.9100{{c}}, ~17/12 = 605.3107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3556{{c}}
 
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
 
Badness (Sintel): 1.42
 
=== 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
 
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 }}
 
Optimal tunings:
* WE: ~2 = 1199.9087{{c}}, ~17/12 = 605.3101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3559{{c}}
 
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
 
Badness (Sintel): 1.56
 
=== 41-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
 
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666
 
Mapping: {{mapping| 1 -8 -34 -32 8 -19 -6 -24 2 21 10 38 -35 | 0 19 72 69 -9 45 20 56 5 -32 -10 -65 80 }}
 
Optimal tunings:
* WE: ~2 = 1199.9179{{c}}, ~17/12 = 605.3156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/12 = 605.3567{{c}}
 
{{Optimal ET sequence|legend=0| 111, 113, 224 }}
 
Badness (Sintel): 1.57
 
== Hemiseptisix ==
Hemiseptisix tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma) in the 7-limit, and may be described as the {{nowrap| 103 & 121 }} temperament. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix. Hemiseptisix was named by [[Xenllium]] in 2021; the name ''hemiseptisix'' refers to a half of septimal major sixth ([[12/7]]).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 95703125/95551488
 
{{Mapping|legend=1| 1 -19 -7 -17 | 0 53 24 51 }}
: mapping generators: ~2, ~98/75
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2693{{c}}, ~98/75 = 466.0801{{c}}
: [[error map]]: {{val| +0.023 -0.149 -0.553 +0.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 466.0715{{c}}
: error map: {{val| 0.000 -0.167 -0.598 +0.819 }}
 
{{Optimal ET sequence|legend=1| 103, 121, 224 }}
 
[[Badness]] (Sintel): 4.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 2734375/2725888
 
Mapping: {{mapping| 1 -19 -7 -17 -28 | 0 53 24 51 81 }}
 
Optimal tunings:
* WE: ~2 = 1200.0183{{c}}, ~55/42 = 466.0767{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0699{{c}}
 
{{Optimal ET sequence|legend=0| 103, 121, 224 }}
 
Badness (Sintel): 1.43
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
 
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 | 0 53 24 51 81 43 }}
 
Optimal tunings:
* WE: ~2 = 1199.9784{{c}}, ~55/42 = 466.0622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/42 = 466.0703{{c}}
 
{{Optimal ET sequence|legend=0| 103, 121, 224 }}
 
Badness (Sintel): 0.873
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
 
Mapping: {{mapping| 1 -19 -7 -17 -28 -13 -13 | 0 53 24 51 81 43 44 }}
 
Optimal tunings:
* WE: ~2 = 1199.8544{{c}}, ~17/13 = 466.0174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 466.0718{{c}}
 
{{Optimal ET sequence|legend=0| 103, 121, 224 }}
 
Badness (Sintel): 0.948
 
== Browser ==
{{See also| Sensipent family }}
 
Named by [[Xenllium]] in 2022, browser may be described as the {{nowrap| 103 & 111 }} temperament.
 
This can also be considered a [[non-over-1 temperament]], with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 78732/78125
 
{{Mapping|legend=1| 1 -29 -37 -47 | 0 35 45 57 }}
: mapping generators: ~2, ~90/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9313{{c}}, ~90/49 = 1048.5414{{c}}
: [[error map]]: {{val| -0.069 -1.013 +0.592 +1.264 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~90/49 = 1048.5998{{c}}
: error map: {{val| 0.000 -0.962 +0.677 +1.362 }}
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
[[Badness]] (Sintel): 4.57
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 78732/78125
 
Mapping: {{mapping| 1 -29 -37 -47 -28 | 0 35 45 57 36 }}
 
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}
 
{{Optimal ET sequence|legend=0| 103, 214 }}
 
Badness (Sintel): 1.91
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845, 1375/1372
 
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 | 0 35 45 57 36 42 }}
 
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
 
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}
 
Optimal tunings:
* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.04
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
 
Mapping: {{mapping| 1 -29 -37 -47 -28 -33 -23 -91 | 0 35 45 57 36 42 31 109 }}
 
Optimal tunings:
* WE: ~2 = 1199.9145{{c}}, ~11/6 = 1048.5290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6021{{c}}
 
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}
 
Badness (Sintel): 1.07
 
== Grazer ==
Named by [[Xenllium]] in 2022, grazer may be described as the {{nowrap| 113 & 121 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 1071875/1062882
 
{{Mapping|legend=1| 1 -3 -4 -5 | 0 37 51 63 }}
: mapping generators: ~2, ~49/45
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0310{{c}}, ~49/45 = 148.7229{{c}}
: [[error map]]: {{val| +0.031 +0.700 -1.561 +0.563 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/45 = 148.7198{{c}}
: error map: {{val| 0.000 +0.676 -1.606 +0.519 }}
 
{{Optimal ET sequence|legend=1| 113, 121, 234 }}
 
[[Badness]] (Sintel): 5.50
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 218750/216513
 
Mapping: {{mapping| 1 -3 -4 -5 -1 | 0 37 51 63 36 }}
 
Optimal tunings:
* WE: ~2 = 1199.7242{{c}}, ~12/11 = 148.6946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234 }}
 
Badness (Sintel): 2.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 540/539, 2200/2197
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 | 0 37 51 63 36 46 }}
 
Optimal tunings:
* WE: ~2 = 1199.7257{{c}}, ~12/11 = 148.6947{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7230{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234 }}
 
Badness (Sintel): 1.50
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 | 0 37 51 63 36 46 33 }}
 
Optimal tunings:
* WE: ~2 = 1199.5690{{c}}, ~12/11 = 148.6815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7267{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234g }}
 
Badness (Sintel): 1.29
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
 
Mapping: {{mapping| 1 -3 -4 -5 -1 -2 0 4 | 0 37 51 63 36 46 33 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.7269{{c}}, ~12/11 = 148.6928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 148.7227{{c}}
 
{{Optimal ET sequence|legend=0| 113, 121, 234g }}
 
Badness (Sintel): 1.37
 
[[Category:Temperament clans]]
[[Category:Mirkwai clan| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Canopic_clan"